Random Walk with an Excluded Origin

Abstract

The mean square end‐to‐end distance $R_N^2$ is calculated for the subset of all random walk configurations on a D‐dimensional simple cubic lattice which do not return to the starting point. Explicit results are obtained in the limit N ≫ 1 for the one‐, two‐, and three‐dimensional lattices. The values of the first two terms in the asymptotic series for $R_N^2$ are, respectively, N + N, N + N/log N, and N + 0.435/N^−½. An unexpected relation is obtained between $R_N^2$ and S_N, the average number of different lattice sites visited in an N‐step random walk on a perfect lattice. It is $R_N^2{\mathrm{=S}}_N{\mathrm{(S}}_{\text{N+1}}{\mathrm{-S}}_N{)}^{\text{−1}}$ .